Introduction

In recent times, significant progress in the field of graphene research has presented novel prospects for the development of THz sources. Graphene exhibits exceptional electronic and optical properties [1]. Numerous potential applications of graphene have been successfully demonstrated [2,3,4]. Nevertheless, the application of this technology in generating THz radiation sources remains insufficient. There is no doubt that both theoretical and experimental research have shown that the graphene sheet has the amazing ability to keep surface plasmons (SPs) moving in the frequency range between THz and mid-infrared [5, 6]. Graphene SPs (GSPs) demonstrate notable characteristics, including enhanced mode confinement and reduced propagation loss in comparison to SPs observed in noble metal films [7, 8]. Furthermore, adjusting the electrostatic gate voltage or adding chemical doping can modify the features of graphene SPs [9, 10].

Srivastav and Panawr [11, 12] reported THz SPs generation on rippled graphene surface by p-polarized laser beam. Kumar et al. [13] observed Smith-Purcell THz radiation generation over a metallic grating by laser-modulated electron beam. Kumar et al. [14] investigated the excitation of THz radiation in an axially modulated magnetized plasma by nonlinear mixing of two CW-lasers. Kumar and Tripathi [15] studied the Cherenkov THz radiation generation over cylindrical dielectric lined resonator by the electron beam. Srivastav and Panwar [16] discussed Cerenkov THz SPs generation by electron beam on n-Insb in the presence of an external magnetic field. By using a relativistic electron beam, Kumar and Tripathi [17] studied theoretically and numerically SPs generation on a thin plasma cylinder. In their research, Kumar et al. [18] investigated the phenomenon of Cherenkov THz SPs excitation by a relativistic electron beam. An ultrathin metal film was deposited onto a glass substrate for this excitation. Tripathi et al. [19] investigated the intricate realm of Cherenkov THz radiation generation within a waveguide adorned with a dielectric lining featuring a rippled surface. This phenomenon was achieved through the utilization of a relativistic electron beam, with the esteemed researchers disclosing the requisite beam energy to be as high as 2 MeV. Zaho et al. [20] investigated the excitation of THz SPs over a circular cylindrical graphene structure by a cyclotron electron beam and reported that the THz SPs can be tuned by the beam energy and Fermi energy of graphene. Zaho et al. [21] theoretically and numerically reported THz SPs generation on a multi-layer graphene sheet deposited on a dielectric substrate with a buffer layer. Liu and Tripathi [22] investigated how an electron beam and lasers may produce SPs over a metal surface and discovered that metal can excite SPs in the infrared frequency range.

In the present paper, theoretically and numerically investigate Cherenkov THz SPs generation by an electron beam over a graphene surface. Cherenkov THz SPs’s energy may be significantly enhanced. The beam of relativistic electron Cherenkov interaction with SPs results in an altered current charge density and gives rise to the Cherenkov THz SPs. In the “THz GSPs Growth Rate” section, the THz SPs’ growth rate is formulated. Discussion and conclusion are represented in the “Discussion” and “Conclusion” sections, respectively.

Fig. 1
figure 1

(Color Online) An electron beam with thickness \(\Delta\) propagates parallel to the graphene surface at height d

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THz GSPs Growth Rate

Consider an interface between free space and graphene at \(x = 0\) with \(x > 0\) represents free space and \(x \le 0\) represents the graphene surface. Graphene has conductivity \(\sigma _{g} = (i e^{2} E_{F}) (\pi \hbar ^{2} (\omega + i \nu ))\) (if \(\text {E}_{\text {F}} \ge \hbar \omega\)) [23] where e is electron charge, \(\text {E}_{\text {F}}\) is Fermi energy, \(\hbar\) and \(\nu\) Planck constant and collision frequency, respectively.

Let us consider THz GSPs at the interface as follows:

$$\begin{aligned} \textbf{E}_{\omega } = E_{0} e^{- i (\omega t - k_{z} z)}{\left\{ \begin{array}{ll} (\hat{z} + \beta _{1} \hat{x}) e^{- \alpha _{1 } x} &{} \text { for x } > 0 \\ (\hat{z} + \beta _{2} \hat{x}) e^{\alpha _{2 } x} &{} \text { for x} \le 0 \end{array}\right. } \end{aligned}$$
(1)

where \(\alpha ^2_1 = \left( k_{z}^2 - ({\omega ^2}/{c^2}) \right)\), \(\alpha _2^2 = \left( k_{z}^2 - ({\omega ^2}/({c^2} \epsilon _{eff})) \right)\), \(\beta _{1} = -(i k_{z}/\alpha _{1})\), \(\beta _{2} =-\left( { i k_{z}}/{ \alpha _{2} }\right)\), \(\epsilon _{eff} = \epsilon _{\text {SiO}_2} + (i \sigma _{g})/(\omega \epsilon _{0})\), \(\alpha _{1}\) and \(\alpha _2\) are decaying constant, \(k_{z}\) is propagation constant of GSPs wave in direction of \(\hat{z}\).

It follows the dispersion relation as follows [24]:

$$\begin{aligned} \frac{1}{\alpha _{1}} + \frac{\epsilon _{\text {SiO}_2}}{\alpha _{2}} = \frac{\sigma _{g}}{i \omega \epsilon _{0}} \end{aligned}$$
(2)

In Fig. 1, let us assume that the sheet of electron beam density \(n_{0b}\) propagates parallel to the graphene surface at distance d (see Fig. 1). Electron beam density has velocity \(v_{0b}\) in direction \(\hat{z}\) and thickness \(\Delta\) with perturbed density \(n_{0b}=n_{0b}^{0}\Delta \delta (x-d)\) and beam current \(I_{b} = \sqrt{\pi } n_{0b}^{0}\Delta b e v_{b}\) [18], where b denotes the beam width in \(\hat{y}\) direction.

Cherenkov interaction condition for electron beams and GSPs needs \(\omega =k_{z}v_{0b}\) and gives the relationship as

$$\begin{aligned} \frac{1}{\sqrt{\frac{c^{2}}{v_{ob}^{2}-\epsilon _{eff}}}}+\frac{\epsilon _{eff}}{\sqrt{\frac{c^{2}}{v_{ob}^{2}-1}}}=\frac{c \sigma _{g}}{i \epsilon _{0}} \end{aligned}$$
(3)

where \(\gamma _{0b} = (1-v_{0b}^{2}/c^{2})^{-1/2}\).

Fig. 2
figure 2

(Color Online) \(\gamma _{0b}\) Vs THz frequency \(\omega\) for the various values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV (orange line), 0.40 eV (blue dot dashed line) and 0.45 eV (green dashed line)

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Figure 2 represents the \(\gamma _{0b}\) variation with THz frequency \(\omega\) for the various values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV, 0.40 eV and 0.45 eV. The required beam energy (\(E_{b} = (\gamma _{0b}-1)m_{0}c^{2}\)) is higher for excitation at lower THz frequencies \(\omega\). Figure 2 shows that the required \(\gamma _{0b}\) is 1.97469 (beam energy 498.91 KeV) to 1.97492 (499.03 KeV) for the generation of THz frequencies 3 to 10 THz.

The interaction between an electron beam and GSPs wave can be mathematically described using the equation of motion as

$$\begin{aligned} m_{e}^{*}\Bigg [\frac{\partial }{\partial t}(\gamma \textbf{v})+\textbf{v}\cdot (\nabla \gamma \textbf{v})\Bigg ]=-e (\textbf{E}+\textbf{v}\times \textbf{B}) \end{aligned}$$
(4)

Upon solving Eq. (4) with \(\textbf{v} = v_{0b}\hat{z} + \textbf{v}_{1b}\) and subsequently linearizing the equation, we obtain the perturbed velocity of the beam

$$\begin{aligned} v^z_{1 b} = \frac{e}{\gamma ^3_{0 b} m_{e}^{*} i \{ \omega - k_z v_{0 b} \}} E_z \end{aligned}$$
(5)

By employing the continuity equation \({\partial n/\partial t} + \mathbf {\nabla } \cdot (n \textbf{v}) = 0\), where \(n = n_{0b}+n_{1}\), it is possible to derive the perturbed charge density \(n_{1}\) as

$$\begin{aligned} n_1 = \frac{n_{0 b} }{(\omega - k_z v_{0 b})^2 } \frac{e k_z}{m_{e}^{*} i} \frac{E_z}{\gamma ^3_{0 b}} \end{aligned}$$
(6)

The perturbed current density \(\textbf{J}_{\omega } = n q \textbf{v}\) becomes

$$\begin{aligned} J^{\omega }_{1, z} = \frac{i \triangle }{(\omega - k_z v_{0 b})^2 } \frac{n^0_{0 b} e^2 }{m_{e}^{*}} \frac{\omega }{\gamma ^3_{0 b} } E_{0} \delta (x - d) e^{\alpha _2 x} e^{- i (\omega t - k_z z)} \end{aligned}$$
(7)

In the absence of an electron beam, the GSPs wave field mode structures are specified as \(\textbf{E}_{s}\) and \(\textbf{H}_{s}\)

$$\begin{aligned} \mathbf {\nabla } \times \textbf{E}_s = - \mu _0 \frac{\partial \textbf{H}_s}{\partial t}\quad \text {and}\quad \mathbf {\nabla } \times \textbf{H}_s = - i \omega \epsilon _0 \epsilon _{eff} \textbf{E}_s \end{aligned}$$
(8)

In the presence of the perturbed beam current, the electric and magnetic fields can be expressed in terms of the electric and magnetic fields of the GSPs wave as [18, 25]

$$\begin{aligned} \textbf{E}_b = E_1 (t) \textbf{E}_s \quad \text {and}\quad \textbf{H}_b = E_2 (t) \textbf{H}_s \end{aligned}$$
(9)

Solving Faraday’s law \(\mathbf {\nabla } \times \textbf{E}_b = - \mu _0 (\partial \textbf{H}_b/\partial t)\) and Ampere’s law, \(\mathbf {\nabla }\times \textbf{H}_b = J^{\omega }_{1,z}\hat{z} + (\partial \textbf{D}_b/\partial t)\) with Eq. (9), we get

$$\begin{aligned} \frac{\partial E_2}{\partial t} = i \omega (E_2 - E_1) \end{aligned}$$
(10)

and

$$\begin{aligned} \textbf{E}_s \frac{\partial A_1}{\partial t} = \frac{\textbf{J}^{n l}_{\omega }}{\epsilon _0 \epsilon _{eff}} + i \omega (E_2 - E_1) \textbf{E}_s \end{aligned}$$
(11)

Simplify Eqs. (10) and (11) and multiply by \(\textbf{E}_{s}^{*}\), and integrating over \(-\infty\) to \(\infty\), we obtain

$$\begin{aligned} \frac{\partial E_1}{\partial t} = \frac{I_2}{\epsilon _0 \epsilon _{eff} I_1} \end{aligned}$$
(12)

where \(I_1 = \int ^{\infty }_{- \infty } \textbf{E}_s \cdot \textbf{E}_s^{*} d x\) and \(I_{2} = \int ^{\infty }_{- \infty } J^{\omega }_{1, z} \cdot \textbf{E}_{s z}^{*} d x\).

After solving the integrals of above equation and assuming the beam current is present \(A = 1\), one may obtain

$$\begin{aligned} \frac{\partial E_1}{\partial t} =\frac{2}{\epsilon _{eff} \left( \frac{1 + \beta ^2_2}{\alpha _2} + \frac{1 + \beta ^2_1}{\alpha _1} \right) } \frac{i \triangle }{(\omega - k_z \text {v}_{0 b})^2 } \omega ^2_{p b} \frac{\omega }{\gamma ^3_{0 b} } E_1 e^{- 2\alpha _1 d} \end{aligned}$$
(13)

where \(\omega _{pb} = (n_{0b}^{0} e^{2}/m_{e}^{*} \epsilon _{0})^{1/2}\) is plasma frequency of beam.

Let \(\omega - k_z \text {v}_{0 b} = i{\Gamma }\) and \(\partial /\partial t \sim {\Gamma }\), we obtain,

$$\begin{aligned} \Gamma ^{3}= R e^{- i(\frac{2l-1}{2}) \pi } \end{aligned}$$

where

$$\begin{aligned} R = \frac{2 \triangle }{\epsilon _{eff} \left( \frac{1 + \beta ^2_2}{\alpha _2} + \frac{1 + \beta ^2_1}{\alpha _1} \right) } \omega ^2_{p b} \frac{\omega }{\gamma ^3_{0 b} } e^{- 2\alpha _1 d} \end{aligned}$$

Or,

$$\begin{aligned} \Gamma = R^{1/3} e^{- i(\frac{2l-1}{2}) \frac{\pi }{3} } \end{aligned}$$
(14)

and real part of THz GSPs’ growth rate (\(\Gamma _{r}\)) is given by for \(l=1\)

$$\begin{aligned} \Gamma _{r}= \frac{\sqrt{3}}{2} \left( \frac{2 \triangle }{\epsilon _{eff} \left( \frac{1 + \beta ^2_2}{\alpha _2} + \frac{1 + \beta ^2_1}{\alpha _1} \right) } \omega ^2_{p b} \frac{\omega }{\gamma ^3_{0 b} } e^{- 2\alpha _1 d} \right) ^{1 / 3} \end{aligned}$$
(15)

THz GSPs’ growth rate depends on the beam density \(n_{0b}\) and Fermi energy of the graphene surface \(\text {E}_{\text {F}}\).

The THz GSPs’efficiency [25] may be taken as

$$\begin{aligned} \eta =\frac{\Gamma _{r} \gamma _{0b}^{3}}{\omega (\gamma _{0b}-1)} \end{aligned}$$
(16)

Using the value of \(\Gamma _{r}\) from Eq. (15) in Eq. (16), the efficiency of THz GSPs becomes

$$\begin{aligned} \eta =\frac{\sqrt{3}}{2} \left( \frac{2 \triangle \omega ^2_{pb} \omega }{\epsilon _{eff} \left( \frac{1 + \beta ^2_2}{\alpha _2} + \frac{1 + \beta ^2_1}{\alpha _1} \right) } e^{- 2\alpha _1 d} \right) ^{1 / 3}\frac{\gamma _{0b}^{2}}{\omega (\gamma _{0b}-1)} \end{aligned}$$
(17)

Equation (17) represents the efficiency of THz GSPs’ growth rate and it is directly proportional to the cube root of \(n_{0b}\).

The following parameters \(\text {d} = 6.7 \times 10^{2} \mu m\), \(\Delta = 38.7 \times 10^{2} \mu m\), beam density \(n_{0b} = 5.72 \times 10^{16} \text {m}^{-3}\), \(\gamma _{0b} = 1.97469\) (beam energy \(E_{b} = 498.91\) KeV) and \(\text {c} = 3\times 10^{8} m/sec\) are used for numerical study of THz GSPs’ growth rate and efficiency in Figs. 3, 4, 5, and 6. Figures 3 and 4 represent the variation of the ratio of THz GSPs’ growth rate (\(\Gamma _{r}^{'}=\Gamma _{r}/\omega\)) with THz frequency \(\omega\).

Fig. 3
figure 3

(Color Online) THz GSPs’ growth rate \(\Gamma _{r}^{'}\) Vs THz frequency \(\omega\) for various values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV (green line), 0.40 eV (red dashed line) and 0.45 eV (blue dot-dashed line) at beam energy \(E_{b} = 498.91\) KeV. The other parameters are \(\text {d} = 6.7 \times 10^{2} \mu m\), \(\Delta = 38.7 \times 10^{2} \mu m\), beam density \(n_{0b} = 5.72 \times 10^{16} \text {m}^{-3}\) and \(\text {c} = 3\times 10^{8} m/sec\)

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Fig. 4
figure 4

(Color Online) THz GSPs’ growth rate \(\Gamma _{r}^{'}\) Vs THz frequency \(\omega\) for various values of beam density \(n_{0b} = 5.72\times 10^{16} m^{-3}\) (green line), \(4.82\times 10^{16} m^{-3}\) (red dashed line) and \(4.32\times 10^{16} m^{-3}\) (blue dot dashed line) at Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV. The other parameters are \(\text {d} = 6.7 \times 10^{2} \mu m\), \(\Delta = 38.7 \times 10^{2} \mu m\) and \(\text {c} = 3\times 10^{8} m/sec\)

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Fig. 5
figure 5

(Color Online) Efficiency of THz GSPs \(\eta\) Vs THz frequency \(\omega\) for various values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV (green line), 0.40 eV (red dashed line) and 0.45 eV (blue dot-dashed line) at beam energy \(E_{b} = 498.91\) KeV. The other parameters are \(\text {d} = 6.7 \times 10^{2} \mu m\), \(\Delta = 38.7 \times 10^{2} \mu m\), beam density \(n_{0b} = 5.72 \times 10^{16} \text {m}^{-3}\) and \(\text {c} = 3\times 10^{8} m/sec\)

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Fig. 6
figure 6

(Color Online) Efficiency of THz GSPs \(\eta\) Vs THz frequency \(\omega\) for various values of beam density \(n_{0b} = 5.72\times 10^{16} m^{-3}\) (green line), \(4.82\times 10^{16} m^{-3}\) (red dashed line) and \(4.32\times 10^{16} m^{-3}\) (blue dot dashed line) at Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV. The other parameters are \(\text {d} = 6.7 \times 10^{2} \mu m\), \(\Delta = 38.7 \times 10^{2} \mu m\) and \(\text {c} = 3\times 10^{8} m/sec\)

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Discussion

Figure 3 shows the variation of THz GSPs’ growth rate \(\Gamma _{r}^{'}\) with THz frequency \(\omega\) for three values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV, 0.40 eV, and 0.45 eV at beam energy \(E_{b} = 498.91\). THz GSPs’ growth rate exhibits a linear relationship with the THz frequency \(\omega\), reaching its maximum value before declining with further increases in THz frequency. As the Fermi energy of graphene surface \(\text {E}_{\text {F}}\) increases, the THz GSPs’ growth rate peak shifts slightly toward the higher THz frequency. Also, THz GSPs’ growth rate increases with decreases in Fermi energy of graphene surface \(\text {E}_{\text {F}}\). The variation of THz GSPs’ growth rate \(\Gamma _{r}^{/}\) with THz frequency \(\omega\) for the three values of beam density \(n_{0b} = 5.72\times 10^{16} m^{-3}\), \(4.82\times 10^{16} m^{-3}\), and \(4.32\times 10^{16} m^{-3}\) at Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV are plotted in Fig. 4. THz GSPs’ growth rate rises as the THz frequency \(\omega\) increases, reaching its maximum value before declining as the THz frequency rises further. THz GSPs’ growth rate rises as beam density \(n_{0b}\) rises. Figure 5 shows the THz GSPs’ efficiency \(\eta\) with THz frequency for various values of graphene Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV, 0.40eV, and 0.45 eV at beam energy \(E_{b} = 498.91\) KeV. When THz frequency \(\omega\) increases, the efficiency of THz GSPs increases to a maximum and then decreases with additional THz frequency increases. THz GSPs’ efficiency increases with decreases in Fermi energy of graphene surface \(\text {E}_{\text {F}}\). Figure 6 shows the efficiency of THz GSPs’ growth rate \(\eta\) with THz frequency for various values of beam density \(n_{0b} = 5.72\times 10^{16} m^{-3}\) (green line), \(4.82\times 10^{16} m^{-3}\) (red dashed line), and \(4.32\times 10^{16} m^{-3}\) (blue dot dashed line) at Fermi energy \(\text {E}_{\text {F}} = 0.35\) eV. As the normalized THz frequency increases further, the efficiency of THz GSPs decreases after reaching its maximum efficiency The efficiency of THz GSPs rises as beam density \(n_{0b}\) does.

Conclusion

In the present paper, analytically and numerically studied the THz GSPs generation by Cerenkov interaction using a relativistic electron beam over a graphene surface. The generation of THz GSPs within the frequency range of 3–10 THz necessitates an electron beam energy ranging from 498.91 to 499.03 KeV. Additionally, the Fermi energy (\(\text {E}_{\text {F}}\)) of graphene exhibits variations between 0.35 and 0.45 eV. THz GSPs’ growth rate grows linearly with THz frequency for different Fermi energy of graphene, reaches its maximum value, and then drops further with a THz frequency. THz GSPs’ growth rate rises with a rise in electron beam density. Kumar et al. [18] observed the maximum growth rate of THz SPs over ultra-thin metal film deposited on glass substrate approximately \(10^{-6}\) by the relativistic electron beam. Sharma et al. [25] reported the efficiency of THz SPs of the order of \(10^{-4}\) Here, the growth rate and efficiency of THz GSPs reported of the order of \(10^{-4}\). Graphene Fermi energy, electron beam density, and electron beam energy strongly modify the growth rate and efficiency of THz GSPs. The Fermi energy of a graphene sheet may be adjusted by applying a gate voltage [26], rendering the proposed approach a dynamically adjustable device suitable for a diverse array of applications in THz graphene plasmonic and photonics devices, ultrafast switching, high-speed data transfer, and frequency conversion [27,28,29].